Advances in Mathematical Physics

Advances in Mathematical Physics / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 9860471 | 12 pages |

Numerical Treatment for the Three-Dimensional Eyring-Powell Fluid Flow over a Stretching Sheet with Velocity Slip and Activation Energy

Academic Editor: Ivan Giorgio
Received13 Mar 2019
Accepted18 Apr 2019
Published06 May 2019


In this manuscript, a computational paradigm of technique shooting is exploited for investigation of the three-dimensional Eyring-Powell fluid with activation energy over a stretching sheet with slip arising in the field of fluid dynamics. The problem is modeled and resulting nonlinear system of PDEs is transformed into nonlinear system of ODEs using well-known similarity transformations. The strength of shooting based computing approach is employed to analyze the dynamics of the system. The proposed technique is well-designed for different scenarios of the system based on three-dimensional non-Newtonian fluid with activation energy over a stretching sheet. Slip condition is also incorporated to enhance the physical and dynamical analysis of the system. The proposed results are compared with the bvp4C method for the correctness of the solver. Graphical and numerical illustrations are used to envisage the behavior of different proficient physical parameters of interest including magnetic parameter, stretching rate parameter, velocity slip parameter, Biot number on velocity, and Lewis number on temperature and concentration.

1. Introduction

From the last few decades, considering the non-Newtonian nature of different physiological fluid and their use in the industry, researchers are paying their attention in the fluid mechanics field on non-Newtonian fluid models. Examples of this category of fluids are Reiner-Philippoff fluid, Casson fluid, micropolar fluid, Prandtl-Eyring fluid, Carreau fluid, power law fluid, Prandtl fluid, and Eyring-Powell fluid. Powell et al. [1] witnessed that, at moderate stresses, the fluid velocity is exponentially dependent on the stress, but as the stress level is enhanced, velocity behaves linearly in the flow. Patel et al. [2] reported that the Eyring-Powell model is useful and has significant benefits in comparison with the power law model. Fluid flow close to a dynamic plate for the Eyring-Powell with the help of three different techniques has been studied by Sirohi et al. [3]. Considering the complexity of the fluid time scale with the aid of Powell-Eyring fluid model, the impact of the very small and large shear rate viscosities is reported by Yoon et al. [4]. Nadeem et al. [5] explained the peristaltic flow by using Eyring-Powell fluid in their study. They concluded that, by enhancing the non-Newtonian Eyring-Powell flow parameter, increase in the peristaltic pumping region is observed. Time independent flow of incompressible boundary layer flow of Eyring-Powell nanofluid past shrinking sheet has been investigated by Motsa et al. [6]. Jayachandra et al. [7] and later Hayat et al. [8] analyzed the double stratification effects using Eyring-Powell.

The field of nanotechnology has gotten much attention among the engineers and scientists due to its vast applications in industrial and medical sciences like ceramics and drug delivery, etc. Nanofluids are modeled because of the suspension of nanoshaped elements in the base liquid. Most commonly used nanoparticles are , , , , and their oxides. The nanofluid flow passing through stretching sheet near stagnation point was analyzed by Mustafa et al. [9]. Makinde et al. [10] analyzed nonisothermal boundary layer nanofluid fluid because of linear stretching surface. Sheikholeslami et al. [11] reported the heat transfer properties for nanofluid flow in the channel. Goodarzi et al. [12] reported nanofluids flow for laminar and turbulent case in a channel of narrow cavity, by using two types mixture model. Malvandi et al. [13] presented the flow for multiple convection nanofluid underneath the transverse narrow channel by using modified Buongiorno’s model. Kuznetsov et al. [14] reported natural convection nanofluid flow past a vertical plate. The 3D flow of second grade nanofluid was analyzed by Hayat et al. [15]; they used an exponentially stretching surface and studied the flow due to the thermal radiation effect and heat source/sink. Hedayati et al. [16] analyzed the convection enforced nanofluid flow in a narrow channels using asymmetrically provided heat. Using the heat source/sink, three-dimensional Maxwell nanofluid boundary layer flow was studied by Hayat et al. [17]. Ellahi et al. [18] discussed entropy induced nanofluid flow using the behaviors of nanoparticles geometry. Latiff et al. [19] reported the bioconvection boundary layer micropolar nanofluid flow over a variable size sheet using diverse slip impacts. Uddin et al. [20] analyzed bioconvection nanofluid flow past a dynamic plate using the variations of Stefan injection along with diverse slip.

Several models for the slip at the boundary flow have been investigated in the span of previous few decades. Some intensively employed models are Maxwell’s boundary-slip flow model of order first [21], slip boundary flow model of second order [22], Fukui-Kaneko (FK) slip boundary flow model [23], and slip boundary flow model of fractional order [24]. All of these models have been originated with the help of kinetic theory for gases. The Fukui-Kaneko slip model is adequately precise; however, it very much relies on the shape of the flow and dependent upon the flow parameters involved. These velocity slip models of first grade, second grade, and fractional grade work well when the Kundsen number is taken between zero and one. The aforementioned models are extensively being employed in science and engineering problems. Currently, Lin Wu [25] studied enhanced slip model of second grade that is considered accurate and reliable in comparison with the FK model. This model is considered superior over previous slip models, because of its applicability for all Kundsen number. For more insightful understanding of the flow behaviors, several researchers worked on Wu’s model using various physical parameters. Furthermore, Ibrahim and Shankar [26] introduced the slips of three types: velocity, solutes, and thermal, due to flow of MHD nanofluid through a stretching sheet.

Zhang et al. [27] developed a new scheme to optimize the GRNN parameter and also introduced a new nonlinear function fitting scheme. Duan et al. [28] presented incompressible SPH algorithm by soothing the constant invariant density scenario for fluid of constant density. Qing et al. [29] by using finite volume technique explored the impacts of water entry angle through cavitation flow region.

The activation energy is the least required energy that reactants have to obtain, so that a chemical reaction can be anticipated. The phenomenon of mass transfer happens because of concentration variance of types in a mixture. The types that can vary the concentration in a mixture flow from large concentration section to less concentration section. The activation energy has many advantages in chemical engineering industry, food processing, mechanics of water, oil emulsions and geothermal reservoirs, etc. Bestman [30] studied the flow using naturally convected binary mixture in a permeable medium along with activation energy. Makinde et al. [31] reported the flow of natural convection with the impacts of reaction, for order n and activation energy through numerical computations. Maleque [32] examined, with the aid of activation energy, the reactions of exothermic or endothermic on variant convective flows. Hongchun et al. [33] presented meshless barycentric interpolation collocation technique for finding the solution of PDEs which are encountered in many physical problems. Zhang et al. [34] presented numerical technique for fraction control problems using Chebyshev polynomials. Yao et al. [35] developed a mathematical model for investigating the loss of root stone due water flow in dam structure. Hemeda et al. [36] proposed an iterative technique along with integral iterative scheme for linear and nonlinear Fokker-Planck equations. Awad et al. [37] used the time dependent rotating binary fluid flow over a suddenly stretched sheet using the enhanced Arrhenius function. Abbas et al. [38] also studied flow of Casson nanofluid in the locality of stagnation point by using dual chemical reaction and activation energy effects. They modeled mathematically the flow system using spectral-collocation quasi-linearization method. Shafique et al. [39] used numerical approach and give the idea of rotating viscoelastic flow having species of chemically reaction with activation energy.

The novel contributions of the study are presented as follows:(i)A novel investigation is presented for modeling and analyzing numerically by shooting method for the three-dimensional Eyring-Powell fluid system with activation energy over a stretching sheet arising in the field of fluid dynamics(ii)The strength of similarity transformation is exploited to transform the nonlinear PDEs of fluidic model into system of nonlinear ODEs.(iii)The proposed results are compared with the state-of-the-art counterpart to prove its effectiveness and correctness.(iv)Numerical and graphical illustrations are utilized to ascertain the value of the scheme by visualizing and analyzing the performance of proficient physical quantities on velocity, temperature, and concentration profiles of the system.

The rest of the paper is organized as follows: mathematical formulations of the problem are described in Section 2; in Section 3, results of numerical experiments are provided, while the conclusion along with future research studies is provided in the last section.

In the current investigation, the three-dimensional Eyring-Powell fluid flow near a stretching surface with velocity slip and activation energy is discussed. To discover the full insight of the behavior of different parameters on velocities, energy, and concentration utilities, a well-known shooting technique along with fourth-order Runge-Kutta technique has been employed.

2. Mathematical Formulation

In this section, the detailed mathematical analysis for time independent incompressible three-dimensional Eyring-Powell fluid boundary layer flow, in the presence of activation energy, has been investigated. The region that occupies the flow domain is considered. The sheet is extended in two directions with fixed origin. The velocities of the sheet are and along the longitudinal and transverse directions. Coordinate scale and geometrical interpretation of the problem are exhibited in Figure 1. The velocity component is along -axis, is along -axis, and is along -axis. represents nanoparticles concentration at sheet and is showing the ambient concentration. and are representing the temperature of the convective region and ambient surface. The equations of motions along energy and concentration of the Eyring-Powell model with the effects of activation energy can be expressed asThe boundary conditions of the above-mentioned problem are defined asWe define the below similarity transforms:By employing similarity transformations, the nondimensional forms of (1)-(5) are transformed asWith the help of similarity transformation (6) can be described asVarious unitless parameters arising in the aforementioned equations are given asThe local skin friction coefficients can be defined in and directions asNow we can define the skin frictions and in and directions; the thermal and mass heat fluxes are and defined as follows.

The dimensional local Nusselt number and Sherwood number, respectively, can be interpret in the form asUsing (14) to (16), one can easily getHere is the Reynolds number which is relied upon on the stretching velocity.

3. Solution Methodology

The solutions for coupled nonlinear ODEs (8)-(11) with the aid of boundary conditions (12) are determined by employing the famous shooting method. For the determination of numerical solution, the domain of the problem has been taken as instead of . It is observed that the solution for has disregarded variations. The initial value problem is obtained from the above boundary value problems. A system of initial value problem has been transformed, in which by , by , by , and by have been used. The resulting equations are written aswhere , , , and are the missing initial conditions. To update the initial conditions, Newton’s method has been used. The stopping conditions of the iterative process have been set aswhere is a small positive number. In the present study, the numerical outcomes are obtained for

Table 1 shows the deep analysis of presently studied results of and against the Hartmann number and stretching rate ratio parameter with those of Freidoonimehr et al. [40] and Hayat et al. [41]. The computed results in Table 1 show a very good comparison with the published numerical results. Table 2 shows the comparison of the numerical results of and against the stretching rate ratio parameter with those of Wang [42] and Freidoonimehr et al. [40]. A fabulous agreement is noticed between the computed results and the published numerical results in Table 2. For further validation and strengthening of proposed solver, the obtained numerical results have been compared with the MATLAB built-in solver bvp4c.

Ref. [33] Ref. [34] shooting bvp4c Ref. [33] Ref. [34] shooting bvp4c

0 0 1 1 1 1 0 0 0 0
1 0 1.414213 1.414214 1.414213 1.414213 0 0 0 0
0 0.5 1.093095 1.093095 1.093105 1.093105 0.465205 0.465205 0.465213 0.465213
1 0.5 1.476770 1.476771 1.476770 1.476770 0.679809 0.679809 0.679809 0.679809
0 1 1.173721 1.173722 1.173723 1.173723 1.173721 1.173722 1.173723 1.173723
1 1 1.535710 - 1.535710 1.535710 1.535710 - 1.535710 1.535710

Ref. [35] Ref. [33] shooting bvp4c Ref. [35] Ref. [33] shooting bvp4c

0 1 1 1 1 0 0 0 0
0.25 1.048813 1.048812 1.048834 1.048834 0.194564 0.194564 0.194576 0.194576
0.5 1.093097 1.093095 1.093105 1.093105 0.465205 0.465205 0.465213 0.465213
0.75 1.134485 1.134485 1.134491 1.134491 0.794622 0.794622 0.794623 0.794623
1 1.173720 1.173721 1.173723 1.173723 1.173720 1.173720 1.173720 1.173720

4. Analysis of Numerical Results

Figures 25 demonstrate the behavior of the velocity components and , temperature profile , and concentration distribution against the Hartmann number . The drag force famous as Lorentz force is obtained by executing the magnetic field vertically through electrically conducting fluid. It is the capability of Lorentz force to make the flow slow which is passing through stretching sheet. With the enhancement in Hartmann number , velocity profile reduces in the directions of and . As stated above, the magnetic parameter relies upon the Lorentz force which provides resistance to the flow. By enhancing the Hartmann number, the well-known Lorentz force increases, due to this reason decline in all the component of velocity is observed in the fluid. Moreover, along with this behavior, a small enhancement is observed in the temperature along with concentration distribution by rising the magnetic number . This demonstrates that it is very clear that the transport phenomena are opposed by the transverse magnetic field. It would be significant to describe here that the huge resistances on the fluid elements create heat generation in the flow field, due to this magnetic field increases vertically. Figures 69 narrates the velocity slip parameter effects on both of the fluid velocity components, temperature, and concentration profiles. Here in this study, the variation of slip parameter is . By enhancing the values of , decrement is noticed in the fluid velocity components, while the increment is seen in the temperature and concentration profiles. In another sense, the effect of slip becomes stronger, when the smaller quantity of flow is taken outside and it moved ahead in all directions of the flow. By increasing, creates a decline in the saturation of the fixed surface due to the boundary layer in both the axial and transverse directions, which results in decline in the boundary layer thickness of momentum, as reduction in the flow is observed with the enhancement in slip so that the skin friction is further decreased from the wall. As described earlier, like behavior is viewed for temperature profile along with slip effect for the nanoparticle concentration field. The effects stretching parameter on the different components of fluid velocity, profiles of temperature, and concentration are demonstrated in Figures 1013. By the definition of stretching parameter, when shows the case of nonbidirectional stretching sheet, due to this fact, tend to zero and improvement in the two-dimensional case is achieved. Moreover, in the case of , the stretching parameter becomes similar in the axial and transverse directions. It is worth mentioning here that, by increasing, signifies the sheet velocity in direction; that is, increases; or decrements are noticed in direction, that is, . Generally, as enhances from zero, then the lateral surface moves in the direction. In addition to this, increasing in the values of and decreases in the thermal boundary layer thickness are viewed, because boundary layer thickness and concentration profile are reduced. Figures 14 and 15 illustrate the impacts of the Biot number on the temperature profile and concentration fields. The fixed wall temperature is attained by taking higher values of the Biot number. The heat transfer coefficient increases by enhancing the values of the Biot number. Rise in the heat transfer coefficient enhances the temperature profile. Moreover, the concentration profile that is driven by temperature profile increases by increases in the Biot number. The dynamic of concentration profile verses Lewis number parameter is represented in Figure 16. In boundary layer, region represents the comparative contribution of the rate of thermal diffusion to the species diffusion. Lewis number is increased due to the species boundary layer decline and the concentration distribution tends to zero. The behavior of the temperature profile against thermophoresis parameter is presented in Figure 17. It is noticed that if thermophoresis parameter is increased, enhancement in the thermal and concentration boundary layer thicknesses is achieved. The reason is behind the fact that submersion of the nanoparticles improves the thermal conductivity of the fluid; due to this an increment is seen in the temperature profile. This is quite evident from Figure 18 that, by increasing the dimensionless activation energy , increment is seen in the nanoparticle concentration profile. The reason behind this fact is Arrhenius function. This function reduces when the activation energy increases, leading to start the generative chemical reaction. Figure 19 portrays the reaction rate constant versus mass fraction field . It shows that, by increasing results, the mass fraction field reduces. The reason behind this fact is the presence of the huge concentration gradient at the boundary wall.

5. Conclusion

A new numerical approach based on shooting method is designed effectively for the problem arising in the field of fluid dynamics represented with the three-dimensional non-Newtonian Eyring-Powell fluid flow near a stretching surface along with the variations of velocity slip and the Arrhenius activation energy. The potential key inferences are summarized as follows:(i)The strength of the proposed solver has been proved by comparing with the results of state of art solver bvp4c(ii)By increasing the Hartmann number and velocity slip parameter, the velocity profile of the fluid declines, while increment is observed in the temperature distribution and concentration profile.(iii)The activation energy strength enhances the mass fraction field, where as the wall mass flux declines.(iv)Rise in the temperature and concentration profiles is witnessed with the increase of Biot number.

In future one may investigate stochastic numerical paradigm based on evolutionary and swarming optimized spline method for the superior numerical treatment of given fluidic system


:Constant magnetic field
:Material parameter
:Biot number
:Magnetic number
:Lewis number
:Brownian motion parameter
:Thermophoresis parameter
:Nanoparticle concentration
:Concentration of nanoparticle
:Prandtl number
:Ambient concentration
:Friction coefficient
:Auxiliary function
:Coefficient of Brownian diffusion
:Reynolds number
:Coefficient of thermophoretic diffusion
:Kinematic viscosity
:Brownian motion factor
:Activation energy
:Material parameter
:Thermophoresis factor
:Reaction rate constant
:Fitted rate constant
:Velocity similarity functions
:Convective surface temperature
:Ambient temperature
:Components of velocity
:Cartesian coordinates
:Thermophoresis factor
:Thermal diffusivity
:Slip parameter
:Slip length
:Similarity parameter
:Temperature distribution
:Stretching parameter
:Fluid density
:Thermal conductivity
:Nondimensional fluid parameter

Data Availability

All the data used in this research work is available within the manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.


  1. R. E. Powell and H. Eyring, “Mechanisms for the relaxation theory of viscosity,” Nature, vol. 154, no. 3909, pp. 427-428, 1944. View at: Publisher Site | Google Scholar
  2. M. Patel and M. G. Timol, “Numerical treatment of Powell-Eyring fluid flow using method of satisfaction of asymptotic boundary conditions (MSABC),” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2584–2592, 2009. View at: Publisher Site | Google Scholar | MathSciNet
  3. V. Sirohi, M. G. Timol, and N. L. Kalthia, “Powell-Eyring model flow near an accelerated plate,” Fluid Dynamics Research, vol. 2, no. 3, pp. 193–204, 1987. View at: Publisher Site | Google Scholar
  4. H. K. Yoon and A. J. Ghajar, “A note on the Powell-Eyring fluid model,” International Communications in Heat and Mass Transfer, vol. 14, no. 4, pp. 381–390, 1987. View at: Publisher Site | Google Scholar
  5. S. Nadeem, N. S. Akbar, and M. Ali, “Endoscopic effects on the peristaltic flow of an Eyring-Powell fluid,” Meccanica, vol. 47, no. 3, pp. 687–697, 2012. View at: Publisher Site | Google Scholar | MathSciNet
  6. T. M. Agbaje, S. Mondal, S. S. Motsa, and P. Sibanda, “A numerical study of unsteady non-Newtonian Powell-Eyring nanofluid flow over a shrinking sheet with heat generation and thermal radiation,” Alexandria Engineering Journal, vol. 56, no. 1, pp. 81–91, 2017. View at: Publisher Site | Google Scholar
  7. M. J. Babu, N. Sandeep, and C. S. K. Raju, “Heat and mass transfer in MHD eyring-powell nanofluid flow due to cone in porous medium,” International Journal of Engineering Research in Africa, vol. 19, p. 57, 2015. View at: Publisher Site | Google Scholar
  8. T. Hayat, M. Zubair, M. Waqas, A. Alsaedi, and M. Ayub, “On doubly stratified chemically reactive flow of Powell–Eyring liquid subject to non-Fourier heat flux theory,” Results in Physics, vol. 7, pp. 99–106, 2017. View at: Publisher Site | Google Scholar
  9. M. Mustafa, T. Hayat, I. Pop, S. Asghar, and S. Obaidat, “Stagnation-point flow of a nanofluid towards a stretching sheet,” International Journal of Heat and Mass Transfer, vol. 54, no. 25-26, pp. 5588–5594, 2011. View at: Publisher Site | Google Scholar
  10. O. D. Makinde and A. Aziz, “Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition,” International Journal of Thermal Sciences, vol. 50, no. 7, pp. 1326–1332, 2011. View at: Publisher Site | Google Scholar
  11. M. Sheikholeslami and D. D. Ganji, “Heat transfer of Cu-water nanofluid flow between parallel plates,” Powder Technology, vol. 235, pp. 873–879, 2013. View at: Publisher Site | Google Scholar
  12. M. Goodarzi, M. Safaei, K. Vafai et al., “Investigation of nanofluid mixed convection in a shallow cavity using a two-phase mixture model,” International Journal of Thermal Sciences, vol. 75, pp. 204–220, 2014. View at: Publisher Site | Google Scholar
  13. A. Malvandi and D. D. Ganji, “Mixed convective heat transfer of water/alumina nanofluid inside a vertical microchannel,” Powder Technology, vol. 263, pp. 37–44, 2014. View at: Publisher Site | Google Scholar
  14. A. V. Kuznetsov and D. A. Nield, “Natural convective boundary-layer flow of a nanofluid past a vertical plate: a revised model,” International Journal of Thermal Sciences, vol. 77, pp. 126–129, 2014. View at: Publisher Site | Google Scholar
  15. T. Hayat, T. Muhammad, S. A. Shehzad, and A. Alsaedi, “Similarity solution to three dimensional boundary layer flow of second grade nanofluid past a stretching surface with thermal radiation and heat source/sink,” AIP Advances, vol. 5, no. 1, Article ID 017107, 2015. View at: Google Scholar
  16. F. Hedayati, A. Malvandi, M. H. Kaffash, and D. D. Ganji, “Fully developed forced convection of alumina/water nanofluid inside microchannels with asymmetric heating,” Powder Technology, vol. 269, pp. 520–531, 2015. View at: Publisher Site | Google Scholar
  17. T. Hayat, T. Muhammad, S. A. Shehzad, and A. Alsaedi, “Three-dimensional boundary layer flow of Maxwell nanofluid: mathematical model,” Applied Mathematics and Mechanics-English Edition, vol. 36, no. 6, pp. 747–762, 2015. View at: Publisher Site | Google Scholar | MathSciNet
  18. R. Ellahi, M. Hassan, and A. Zeeshan, “Shape effects of nanosize particles in Cu - H 2O nanofluid on entropy generation,” International Journal of Heat and Mass Transfer, vol. 81, pp. 449–456, 2015. View at: Publisher Site | Google Scholar
  19. N. A. Abdul Latiff, M. J. Uddin, O. A. Bég, and A. I. Ismail, “Unsteady forced bioconvection slip flow of a micropolar nanofluid from a stretching/shrinking sheet,” Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanomaterials, Nanoengineering and Nanosystems, vol. 230, no. 4, pp. 177–187, 2016. View at: Publisher Site | Google Scholar
  20. M. J. Uddin, M. N. Kabir, and O. A. Bég, “Computational investigation of Stefan blowing and multiple-slip effects on buoyancy-driven bioconvection nanofluid flow with microorganisms,” International Journal of Heat and Mass Transfer, vol. 95, pp. 116–130, 2016. View at: Publisher Site | Google Scholar
  21. J. Maxwell Clerk, “VII. On stresses in rarified gases arising from inequalities of temperature,” Philosophical Transactions of the royal society of London, vol. 170, pp. 231–256, 1879. View at: Google Scholar
  22. Y-T. Hsia and G. A. Domoto, “An experimental investigation of molecular rarefaction effects in gas lubricated bearings at ultra-low clearances,” Journal of Lubrication Technology, vol. 105, no. 1, pp. 120–129, 1983. View at: Google Scholar
  23. S. Fukui and R. Kaneko, “A database for interpolation of poiseuille flow rates for high knudsen number lubrication problems,” Journal of Tribology, vol. 112, no. 1, pp. 78–83, 1990. View at: Publisher Site | Google Scholar
  24. Y. Mitsuya, “Modified reynolds equation for ultra-thin film gas lubrication using 1.5-Order slip-flow model and considering surface accommodation coefficient,” Journal of Tribology, vol. 115, no. 2, pp. 289–294, 1993. View at: Publisher Site | Google Scholar
  25. L. A. Wu, “A slip model for rarefied gas flows at arbitrary Knudsen number,” Applied Physics Letters, vol. 93, no. 25, Article ID 253103, 2008. View at: Publisher Site | Google Scholar
  26. W. Ibrahim and B. Shankar, “MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with velocity, thermal and solutal slip boundary conditions,” Computers & Fluids, vol. 75, pp. 1–10, 2013. View at: Publisher Site | Google Scholar | MathSciNet
  27. Y. Zhang, J. Niu, and S. Na, “A novel nonlinear function fitting model based on FOA and GRNN,” Mathematical Problems in Engineering, vol. 2019, Article ID 2697317, 10 pages, 2019. View at: Publisher Site | Google Scholar
  28. X. Duan, H. Ren, and H. Li, “Incompressible fluids simulation by relaxing the density-invariant condition in a marine simulator,” Mathematical Problems in Engineering, vol. 2019, Article ID 8971089, 11 pages, 2019. View at: Publisher Site | Google Scholar
  29. Q. Mu, Y. Lv, K. Wang, T. Xiong, and W. Yi, “Numerical simulation on the cavitation flow of high speed oblique water entry of revolution body,” Mathematical Problems in Engineering, vol. 2019, Article ID 8034619, 10 pages, 2019. View at: Publisher Site | Google Scholar | MathSciNet
  30. A. R. Bestman, “Natural convection boundary layer with suction and mass transfer in a porous medium,” International Journal of Energy Research, vol. 14, no. 4, pp. 389–396, 1990. View at: Publisher Site | Google Scholar
  31. O. D. Makinde, P. O. Olanrewaju, and W. M. Charles, “Unsteady convection with chemical reaction and radiative heat transfer past a flat porous plate moving through a binary mixture,” Afrika Matematika, vol. 22, no. 1, pp. 65–78, 2011. View at: Publisher Site | Google Scholar | MathSciNet
  32. K. A. Maleque, “Effects of exothermic/endothermic chemical reactions with Arrhenius activation energy on MHD free convection and mass transfer flow in presence of thermal radiation,” Journal of Thermodynamics, vol. 2013, Article ID 692516, 11 pages, 2013. View at: Google Scholar
  33. H. Wu, Y. Wang, and W. Zhang, “Numerical solution of a class of nonlinear partial differential equations by using barycentric interpolation collocation method,” Mathematical Problems in Engineering, vol. 2018, Article ID 7260346, 10 pages, 2018. View at: Publisher Site | Google Scholar | MathSciNet
  34. J. Zhang, Y. Li, and J. Xie, “Numerical simulation of fractional control system using Chebyshev polynomials,” Mathematical Problems in Engineering, vol. 2018, Article ID 4270764, 5 pages, 2018. View at: Publisher Site | Google Scholar | MathSciNet
  35. C. Yao, C. Yang, Y. Lan, Q. Jiang, S. Jiang, and J. Yang, “Numerical simulation study on root stones loss in dam buttress engineering,” Mathematical Problems in Engineering, vol. 2018, Article ID 5132760, 15 pages, 2018. View at: Publisher Site | Google Scholar
  36. A. A. Hemeda and E. E. Eladdad, “New iterative methods for solving Fokker-Planck equation,” Mathematical Problems in Engineering, vol. 2018, Article ID 6462174, 9 pages, 2018. View at: Publisher Site | Google Scholar | MathSciNet
  37. F. G. Awad, S. Motsa, and M. Khumalo, “Heat and mass transfer in unsteady rotating fluid flow with binary chemical reaction and activation energy,” PLoS ONE, vol. 9, no. 9, Article ID e107622, 2014. View at: Publisher Site | Google Scholar
  38. Z. Abbas, M. Sheikh, and S. S. Motsa, “Numerical solution of binary chemical reaction on stagnation point flow of Casson fluid over a stretching/shrinking sheet with thermal radiation,” Energy, vol. 95, pp. 12–20, 2016. View at: Publisher Site | Google Scholar
  39. Z. Shafique, M. Mustafa, and A. Mushtaq, “Boundary layer flow of Maxwell fluid in rotating frame with binary chemical reaction and activation energy,” Results in Physics, vol. 6, pp. 627–633, 2016. View at: Publisher Site | Google Scholar
  40. N. Freidoonimehr and A. B. Rahimi, “Brownian motion effect on heat transfer of a three-dimensional nanofluid flow over a stretched sheet with velocity slip,” Journal of Thermal Analysis and Calorimetry, pp. 1–16, 2018. View at: Google Scholar
  41. T. Hayat, S. A. Shehzad, M. Qasim, and S. Asghar, “Three-dimensional stretched flow via convective boundary condition and heat generation/absorption,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 24, no. 2, pp. 342–358, 2014. View at: Publisher Site | Google Scholar | MathSciNet
  42. C. Y. Wang, “The three dimensional flow due to a stretching flat surface,” Physics of Fluids, vol. 27, no. 8, pp. 1915–1917, 1984. View at: Publisher Site | Google Scholar | MathSciNet

Copyright © 2019 Muhammad Umar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

More related articles

1125 Views | 473 Downloads | 7 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.